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Boundary states and broken bulk symmetries in W A(r) minimal models

We study the boundary states of (p', p) rational conformal field theories having a W symmetry of the type A(r) using the multi-component free-field formalism. The classification of primary fields for these models given in the literature is shown to be incomplete; we give the correct classification by demanding modular covariance and show that the resulting modular S matrix satisfies all the necessary conditions. Basis states satisfying the boundary conditions are found in the form of coherent states and as expected we find that W violating states can be found for all these models. We construct consistent physical boundary for all the rank 2 (p+1, p) models (of which the already known case of the 3-state Potts model is the simplest example) and find that the W violating sector possesses a direct analogue of the Verlinde formula.